{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T02:31:35Z","timestamp":1769221895832,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,23]],"date-time":"2022-08-23T00:00:00Z","timestamp":1661212800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51479137"],"award-info":[{"award-number":["51479137"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Soft foundation consolidation engineering applications and experimental studies have proven the effectiveness of the air\u2013boosted vacuum preloading method (AVP). The sand well consolidation theory, a typical axisymmetric consolidation, is adapted to analyze the whole consolidation process of the saturated soil by air\u2013boosted vacuum preloading. The present analytical solution for air\u2013boosted vacuum consolidation of the saturated soil is more suitable for application in a deep soft foundation. With the solving method, the general solution through separation of variables is used, which is mathematically used for the homogeneous partial differential equations. However, the partial differential equations for solving the consolidation of the AVP method are nonhomogeneous. Therefore, the eigenfunction expansion method for nonhomogeneous equations is proposed and validated in this study. Results showed that the improved analytical solution by the eigenfunction expansion method is more consistent with the numerical solution than that of the previous method using the general solution by separation of variables, which leads to a lower error ratio of less than 2%. The improved analytical solution can be used to predict the consolidation of deep foundations by air\u2013boosted vacuum preloading effectively.<\/jats:p>","DOI":"10.3390\/sym14091757","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T02:55:34Z","timestamp":1661309734000},"page":"1757","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Improved Analytical Solution for Air\u2013Boosted Vacuum Consolidation of Saturated Soil Using Eigenfunction Expansion Method"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8287-9706","authenticated-orcid":false,"given":"Liquan","family":"Xie","sequence":"first","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8011-8572","authenticated-orcid":false,"given":"Zelong","family":"Liang","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"given":"Guangrui","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, Tongji University, Shanghai 200092, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2579-0526","authenticated-orcid":false,"given":"Yanhong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4232-7784","authenticated-orcid":false,"given":"Tongqing","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Architecture, Chongqing University of Science and Technology, Chongqing 401331, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,23]]},"reference":[{"key":"ref_1","first-page":"73","article-title":"Soil Reinforcement Using Coconut Shell Ash: A Case Study of Indian Soil","volume":"6","author":"Prasanna","year":"2017","journal-title":"J. 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